Question

one method used to measure the speed of supercomputers is the number of floating - point mathematical operations the computer can perform in one second. this is often referred to by the acronym flops. for many years since 1992, the number of flops performed by the largest supercomputer available that year was recorded. based on the three graphs shown, which type of model is most appropriate for comparing years to flops? a linear model, because the scatterplot of years and in operations is roughly linear. a power model, because the scatterplot of years and operations shows a steep curve. an exponential model could be appropriate because the scatterplot of years and ln(operations) is roughly linear. the next step is to look at the residual plot. a power model, because the scatterplot of ln(years) and ln(operations) shows the strongest linear relationship.

Explanation:

• For an exponential model \(y = ab^{x}\), taking the natural - logarithm of both sides gives \(\ln y=\ln a + x\ln b\). If the scatter - plot of \(x\) (years) and \(\ln y\) (ln(operations)) is roughly linear, it is an indication that an exponential model might be appropriate.
• A linear model for \(y\) (operations) and \(x\) (years) is not appropriate because the original scatter - plot of years and operations is not linear.
• A power model has the form \(y = ax^{b}\), and taking the natural - logarithm of both sides gives \(\ln y=\ln a + b\ln x\). There is no indication from the problem description that the relationship between \(\ln(\text{years})\) and \(\ln(\text{operations})\) is relevant. Also, the steep curve in the years - operations plot is more indicative of an exponential (\(y = ab^{x}\)) rather than a power (\(y = ax^{b}\)) relationship when considering the transformation for linearity.

Answer:

An exponential model could be appropriate because the scatterplot of years and ln(operations) is roughly linear. The next step is to look at the residual plot.